One more formula for the determinant of Laplacian on a compact Riemann surface

Let $f$ be a meromorphic function on a compact Riemann surface $X$ and let $m$ be the standard roundmetric of curvature $1$ on the Riemann sphere. Then the pullback $f^*m$ of $m$ under $f$ is a metric on$X$ of curvature $1$ with conical singularities at the critical points of $f$. We study the $\zeta$-regularizeddeterminant of the Laplace operator on $X$ corresponding to the metric $f^*m$ as a functional on the modulispace of pairs $(X, f)$ and derive an explicit formula for the functional. The talk is based on the joint workwith V. Kalvin (Concordia University).